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<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>week2</title><link href='https://fonts.loli.net/css?family=Open+Sans:400italic,700italic,700,400&subset=latin,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; --title-bar-height:20px; }
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<body class='typora-export os-windows'>
<div id='write'  class=''><div class='md-toc' mdtype='toc'><p class="md-toc-content" role="list"><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#4-多变量线性回归linear-regression-with-multiple-variables">4 多变量线性回归(Linear Regression with Multiple Variables)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#41-多特征multiple-features">4.1 多特征(Multiple Features)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n22"><a class="md-toc-inner" href="#42-多变量梯度下降gradient-descent-for-multiple-variables">4.2 多变量梯度下降(Gradient Descent for Multiple Variables)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n36"><a class="md-toc-inner" href="#43-梯度下降实践1-特征值缩放gradient-descent-in-practice-i---feature-scaling">4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n47"><a class="md-toc-inner" href="#44-梯度下降实践2-学习速率gradient-descent-in-practice-ii---learning-rate">4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n72"><a class="md-toc-inner" href="#45-特征和多项式回归features-and-polynomial-regression">4.5 特征和多项式回归(Features and Polynomial Regression)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n80"><a class="md-toc-inner" href="#46-正规方程normal-equation">4.6 正规方程(Normal Equation)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n124"><a class="md-toc-inner" href="#47-不可逆性正规方程normal-equation-noninvertibility">4.7 不可逆性正规方程(Normal Equation Noninvertibility)</a></span><span role="listitem" class="md-toc-item md-toc-h1" data-ref="n144"><a class="md-toc-inner" href="#5-octavematlab-tutorial">5 Octave/Matlab Tutorial</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n146"><a class="md-toc-inner" href="#51-basic-operations">5.1 Basic Operations</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n147"><a class="md-toc-inner" href="#52-moving-data-around">5.2 Moving Data Around</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n148"><a class="md-toc-inner" href="#53-computing-on-data">5.3 Computing on Data</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n149"><a class="md-toc-inner" href="#54-plotting-data">5.4 Plotting Data</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n150"><a class="md-toc-inner" href="#55-control-statements-for-while-if-statement">5.5 Control Statements: for, while, if statement</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n151"><a class="md-toc-inner" href="#56-向量化vectorization">5.6 向量化(Vectorization)</a></span><span role="listitem" class="md-toc-item md-toc-h2" data-ref="n153"><a class="md-toc-inner" href="#5x-常用函数整理">5.x 常用函数整理</a></span></p></div><h1><a name="4-多变量线性回归linear-regression-with-multiple-variables" class="md-header-anchor"></a><span>4 多变量线性回归(Linear Regression with Multiple Variables)</span></h1><h2><a name="41-多特征multiple-features" class="md-header-anchor"></a><span>4.1 多特征(Multiple Features)</span></h2><p><span>对于一个要度量的对象，一般来说会有不同维度的多个特征。比如之前的房屋价格预测例子中，除了房屋的面积大小，可能还有房屋的年限、房屋的层数等等其他特征：</span></p><p><img src="images/20180107_234509.png" referrerpolicy="no-referrer"></p><p><span>这里由于特征不再只有一个，引入一些新的记号</span></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.394ex" height="1.435ex" viewBox="0 -504.2 600 617.7" role="img" focusable="false" style="vertical-align: -0.264ex;"><defs><path stroke-width="0" id="E137-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E137-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex">n</script><span>: 特征的总数 </span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.405ex" height="2.461ex" viewBox="0 -956.9 1466.1 1059.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E98-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E98-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E98-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 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transform="scale(0.707)" xlink:href="#E98-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E98-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E98-MJMAIN-29" x="733" y="0"></use></g></g></svg></span><script type="math/tex">{x}^{\left( i \right)}</script><span>: 代表样本矩阵中第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.994ex" viewBox="0 -755.9 345 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E27-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> 行，也就是第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.994ex" viewBox="0 -755.9 345 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E27-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> 个训练实例。</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.405ex" height="3.745ex" viewBox="0 -1107.7 1466.1 1612.3" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="0" id="E99-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E99-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E99-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E99-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E99-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E99-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E99-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E99-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E99-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E99-MJMATHI-6A" x="808" y="-429"></use></g></svg></span><script type="math/tex">{x}_{j}^{\left( i \right)}</script><span>: 代表样本矩阵中第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.994ex" viewBox="0 -755.9 345 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E27-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> 行的第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-12 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.028ex;"><defs><path stroke-width="0" id="E61-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E61-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex">j</script><span> 列，也就是第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.801ex" height="1.994ex" viewBox="0 -755.9 345 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E27-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-69" x="0" y="0"></use></g></svg></span><script type="math/tex">i</script><span> 个训练实例的第 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="0.985ex" height="2.461ex" viewBox="-12 -755.9 424 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex; margin-left: -0.028ex;"><defs><path stroke-width="0" id="E61-MJMATHI-6A" d="M297 596Q297 627 318 644T361 661Q378 661 389 651T403 623Q403 595 384 576T340 557Q322 557 310 567T297 596ZM288 376Q288 405 262 405Q240 405 220 393T185 362T161 325T144 293L137 279Q135 278 121 278H107Q101 284 101 286T105 299Q126 348 164 391T252 441Q253 441 260 441T272 442Q296 441 316 432Q341 418 354 401T367 348V332L318 133Q267 -67 264 -75Q246 -125 194 -164T75 -204Q25 -204 7 -183T-12 -137Q-12 -110 7 -91T53 -71Q70 -71 82 -81T95 -112Q95 -148 63 -167Q69 -168 77 -168Q111 -168 139 -140T182 -74L193 -32Q204 11 219 72T251 197T278 308T289 365Q289 372 288 376Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E61-MJMATHI-6A" x="0" y="0"></use></g></svg></span><script type="math/tex">j</script><span> 个特征。</span></p></blockquote><p><span>参照上图，则有 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.16ex" height="12.383ex" viewBox="0 -2917 11694 5331.5" role="img" focusable="false" style="vertical-align: -5.608ex;"><defs><path stroke-width="0" id="E100-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 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xlink:href="#E100-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,362)"><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-29" x="888" y="0"></use></g><use xlink:href="#E100-MJMAIN-3D" x="1575" y="0"></use><g transform="translate(2520,0)"><g transform="translate(0,2850)"><use xlink:href="#E100-MJSZ4-23A1" x="0" y="-1154"></use><g transform="translate(0,-3451) scale(1,2.8272425249169437)"><use xlink:href="#E100-MJSZ4-23A2"></use></g><use xlink:href="#E100-MJSZ4-23A3" x="0" y="-4556"></use></g><g transform="translate(834,0)"><g transform="translate(-15,0)"><g transform="translate(0,2050)"><use xlink:href="#E100-MJMAIN-31"></use><use xlink:href="#E100-MJMAIN-34" x="500" y="0"></use><use xlink:href="#E100-MJMAIN-31" x="1000" y="0"></use><use xlink:href="#E100-MJMAIN-36" x="1500" y="0"></use></g><g transform="translate(625,650)"><use xlink:href="#E100-MJMAIN-33" x="250" y="0"></use></g><g transform="translate(625,-750)"><use xlink:href="#E100-MJMAIN-32" x="250" y="0"></use></g><g transform="translate(375,-2150)"><g transform="translate(250,0)"><use xlink:href="#E100-MJMAIN-34"></use><use xlink:href="#E100-MJMAIN-30" x="500" y="0"></use></g></g></g></g><g transform="translate(2986,2850)"><use xlink:href="#E100-MJSZ4-23A4" x="0" y="-1154"></use><g transform="translate(0,-3451) scale(1,2.8272425249169437)"><use xlink:href="#E100-MJSZ4-23A5"></use></g><use xlink:href="#E100-MJSZ4-23A6" x="0" y="-4556"></use></g></g><use xlink:href="#E100-MJMAIN-2C" x="6340" y="0"></use><g transform="translate(6784,0)"><use xlink:href="#E100-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-32" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-29" x="888" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E100-MJMAIN-31" x="808" y="-434"></use></g><use xlink:href="#E100-MJMAIN-3D" x="8638" y="0"></use><g transform="translate(9693,0)"><use xlink:href="#E100-MJMAIN-31"></use><use xlink:href="#E100-MJMAIN-34" x="500" y="0"></use><use xlink:href="#E100-MJMAIN-31" x="1000" y="0"></use><use xlink:href="#E100-MJMAIN-36" x="1500" y="0"></use></g></g></svg></span><script type="math/tex">{x}^{(2)}\text{=}\begin{bmatrix} 1416\\\ 3\\\ 2\\\ 40 \end{bmatrix}, {x}^{(2)}_{1} = 1416</script></p><p><span>多变量假设函数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.338ex" height="1.994ex" viewBox="0 -755.9 576 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E48-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E48-MJMATHI-68" x="0" y="0"></use></g></svg></span><script type="math/tex">h</script><span> 表示为：</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.396ex" height="2.577ex" viewBox="0 -806.1 16101 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E101-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path stroke-width="0" id="E101-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E101-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E101-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path 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y="0"></use><use xlink:href="#E101-MJMATHI-78" x="389" y="0"></use><use xlink:href="#E101-MJMAIN-29" x="961" y="0"></use></g><use xlink:href="#E101-MJMAIN-3D" x="2802" y="0"></use><g transform="translate(3857,0)"><use xlink:href="#E101-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E101-MJMAIN-2B" x="5002" y="0"></use><g transform="translate(6002,0)"><use xlink:href="#E101-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(6925,0)"><use xlink:href="#E101-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMAIN-31" x="808" y="-213"></use></g><use xlink:href="#E101-MJMAIN-2B" x="8173" y="0"></use><g transform="translate(9173,0)"><use xlink:href="#E101-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10095,0)"><use xlink:href="#E101-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMAIN-32" x="808" y="-213"></use></g><use xlink:href="#E101-MJMAIN-2B" x="11121" y="0"></use><use xlink:href="#E101-MJMAIN-2E" x="11899" y="0"></use><use xlink:href="#E101-MJMAIN-2E" x="12344" y="0"></use><use xlink:href="#E101-MJMAIN-2E" x="12788" y="0"></use><use xlink:href="#E101-MJMAIN-2B" x="13233" y="0"></use><g transform="translate(14011,0)"><use xlink:href="#E101-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMATHI-6E" x="663" y="-213"></use></g><g transform="translate(15004,0)"><use xlink:href="#E101-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E101-MJMATHI-6E" x="808" y="-213"></use></g></g></svg></span><script type="math/tex">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}}+...+{\theta_{n}}{x_{n}}</script></p><p><span>对于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.143ex" height="2.461ex" viewBox="0 -806.1 922.6 1059.4" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E49-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E49-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E49-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E49-MJMAIN-30" x="663" y="-213"></use></g></svg></span><script type="math/tex">\theta_0</script><span>，和单特征中一样，我们将其看作基础数值。例如，房价的基础价格。</span></p><p><span>参数向量的维度为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.394ex" height="2.11ex" viewBox="0 -755.9 2322.4 908.7" role="img" focusable="false" style="vertical-align: -0.355ex;"><defs><path stroke-width="0" id="E106-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E106-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E106-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E106-MJMATHI-6E" x="0" 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x="15055" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-482">h_\theta\left(x\right)=\begin{bmatrix}\theta_0\; \theta_1\; ... \;\theta_n \end{bmatrix}\begin{bmatrix}x_0 \newline x_1 \newline \vdots \newline x_n\end{bmatrix}= \theta^T x</script></div></div><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.478ex" height="2.344ex" viewBox="0 -906.7 1066.8 1009.2" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E105-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 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inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="1.76ex" viewBox="0 -504.6 1025.6 757.9" role="img" focusable="false" style="vertical-align: -0.588ex;"><defs><path stroke-width="0" id="E107-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E107-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 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& \text{Repeat until convergence:} \; \lbrace \\
&{{\theta }_{j}}:={{\theta }_{j}}-\alpha \frac{\partial }{\partial {{\theta }_{j}}}J\left( {\theta_{0}},{\theta_{1}}...{\theta_{n}}  \right) \\
\rbrace
\end{align*}</script></div></div><p><span>解出偏导得：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n28" cid="n28" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-484-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="60.276ex" height="13.083ex" viewBox="0 -3067.8 25952.1 5633" role="img" focusable="false" style="vertical-align: -5.958ex; max-width: 100%;"><defs><path stroke-width="0" id="E494-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 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& \text{repeat until convergence:} \; \lbrace \\
& \theta_j := \theta_j - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_j^{(i)} \; & \text{for j := 0,1...n}\\
\rbrace
\end{align*}</script></div></div><p><span>可展开为：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n30" cid="n30" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-485-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.364ex" height="38.531ex" viewBox="0 -8546.1 19962.2 16589.5" role="img" focusable="false" style="vertical-align: -18.682ex; max-width: 100%;"><defs><path stroke-width="0" id="E495-MJMAIN-72" d="M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 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transform="translate(499,412)"><use transform="scale(0.707)" xlink:href="#E495-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E495-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E495-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E495-MJMAIN-29" x="14565" y="0"></use><use xlink:href="#E495-MJMAIN-22C5" x="15176" y="0"></use><g transform="translate(15677,0)"><use xlink:href="#E495-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,521)"><use transform="scale(0.707)" xlink:href="#E495-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E495-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E495-MJMAIN-29" x="733" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E495-MJMATHI-6E" x="808" y="-211"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-485">\begin{aligned}
& \text{repeat until convergence:} \; \lbrace \\
& \theta_0 := \theta_0 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_0^{(i)}\\
& \theta_1 := \theta_1 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_1^{(i)} \\
& \theta_2 := \theta_2 - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_2^{(i)} \\
& \vdots \\
& \theta_n := \theta_n - \alpha \frac{1}{m} \sum\limits_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) \cdot x_n^{(i)} &\\
\rbrace
\end{aligned}</script></div></div><p><span>当然，同单变量梯度下降一样，计算时需要</span><strong><span>同时更新</span></strong><span>所有参数。</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.766ex" height="2.811ex" viewBox="0 -906.7 5496.7 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E111-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z"></path><path 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y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-486">\theta = \theta - \alpha \frac{1}{m}(X^T(X\theta-y))</script></div></div><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.877ex" viewBox="0 -755.9 852 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E132-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 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class="md-header-anchor"></a><span>4.3 梯度下降实践1-特征值缩放(Gradient Descent in Practice I - Feature Scaling)</span></h2><p><span>在应用梯度下降算法实践时，由于各特征值的范围不一，可能会影响代价函数收敛速度。</span></p><p><span>以房价预测问题为例，这里选取房屋面积大小和房间数量这两个特征。</span></p><p><span>下图中，左图是以原始数据绘制的代价函数轮廓图，右图为采用特征缩放（都除以最大值）后图像。左图中呈现的图像较扁，相对于使用特征缩放方法的右图，梯度下降算法需要更多次的迭代。</span></p><p><img src="images/20180108_100751.png" referrerpolicy="no-referrer"></p><p>&nbsp;</p><p><span>为了优化梯度下降的收敛速度，采用特征缩放的技巧，使各特征值的</span><strong><span>范围尽量一致</span></strong><span>。</span></p><p><span>除了以上图人工选择并除以一个参数的方式，</span><strong><span>均值归一化(Mean normalization)</span></strong><span>方法更为便捷，可采用它来对所有特征值统一缩放：</span></p><p><span> </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="27.442ex" height="4.445ex" viewBox="0 -1208.2 11815.4 1913.9" role="img" focusable="false" style="vertical-align: -1.639ex;"><defs><path stroke-width="0" id="E115-MJMATHI-78" d="M52 289Q59 331 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transform="scale(0.707)" xlink:href="#E115-MJMAIN-2212" x="915" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-61" x="1693" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-76" x="2222" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-65" x="2707" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-72" x="3173" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-61" x="3624" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-67" x="4153" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-65" x="4633" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMAIN-28" x="5099" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-78" x="5488" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMAIN-29" x="6060" y="0"></use></g><g transform="translate(60,-435)"><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-6D" x="0" 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transform="scale(0.707)" xlink:href="#E115-MJMATHI-6E" x="8003" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-69" x="8603" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-6D" x="8948" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-75" x="9826" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-6D" x="10398" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMAIN-28" x="11276" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMATHI-78" x="11665" y="0"></use><use transform="scale(0.707)" xlink:href="#E115-MJMAIN-29" x="12237" y="0"></use></g></g></g></g></svg></span><script type="math/tex">x_i:=\frac{x_i-average(x)}{maximum(x)-minimum(x)}</script><span>, 使得  </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.936ex" height="2.582ex" viewBox="0 -807.7 5139.2 1111.9" role="img" focusable="false" style="vertical-align: -0.706ex;"><defs><path stroke-width="0" id="E116-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E116-MJMATHI-69" d="M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E116-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E116-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 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-225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E116-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E116-MJMATHI-69" x="808" y="-213"></use><use xlink:href="#E116-MJMAIN-2208" x="1193" y="0"></use><use xlink:href="#E116-MJMAIN-28" x="2138" y="0"></use><use xlink:href="#E116-MJMAIN-2212" x="2527" y="0"></use><use xlink:href="#E116-MJMAIN-31" x="3305" y="0"></use><use xlink:href="#E116-MJMAIN-2C" x="3805" y="0"></use><use xlink:href="#E116-MJMAIN-31" x="4250" y="0"></use><use xlink:href="#E116-MJMAIN-29" x="4750" y="0"></use></g></svg></span><script type="math/tex">x_i \in (-1,1)</script></p><p><span>对于特征的范围，并不一定需要使得 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.653ex" height="2.227ex" viewBox="0 -755.9 5017.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E118-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E118-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E118-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 137Q83 142 83 154Q84 164 94 170Z"></path><path stroke-width="0" id="E118-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E118-MJMAIN-2212" x="0" y="0"></use><use xlink:href="#E118-MJMAIN-31" x="778" y="0"></use><use xlink:href="#E118-MJAMS-2A7D" x="1555" y="0"></use><use xlink:href="#E118-MJMATHI-78" x="2611" y="0"></use><use xlink:href="#E118-MJAMS-2A7D" x="3461" y="0"></use><use xlink:href="#E118-MJMAIN-31" x="4517" y="0"></use></g></svg></span><script type="math/tex">-1 \leqslant x \leqslant 1</script><span>，类似于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.846ex" height="2.227ex" viewBox="0 -755.9 4239.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E119-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E119-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 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350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E119-MJMAIN-31" x="0" y="0"></use><use xlink:href="#E119-MJAMS-2A7D" x="777" y="0"></use><use xlink:href="#E119-MJMATHI-78" x="1833" y="0"></use><use xlink:href="#E119-MJAMS-2A7D" x="2683" y="0"></use><use xlink:href="#E119-MJMAIN-33" x="3739" y="0"></use></g></svg></span><script type="math/tex">1\leqslant x \leqslant 3</script><span> 等也是可取的，而诸如 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.298ex" height="2.227ex" viewBox="0 -755.9 7017.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E120-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E120-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E120-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E120-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 137Q83 142 83 154Q84 164 94 170Z"></path><path stroke-width="0" id="E120-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E120-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#E120-MJMAIN-31"></use><use xlink:href="#E120-MJMAIN-30" x="500" y="0"></use><use xlink:href="#E120-MJMAIN-30" x="1000" y="0"></use></g><use xlink:href="#E120-MJAMS-2A7D" x="2555" y="0"></use><use xlink:href="#E120-MJMATHI-78" x="3611" y="0"></use><use xlink:href="#E120-MJAMS-2A7D" x="4461" y="0"></use><g transform="translate(5517,0)"><use xlink:href="#E120-MJMAIN-31"></use><use xlink:href="#E120-MJMAIN-30" x="500" y="0"></use><use xlink:href="#E120-MJMAIN-30" x="1000" y="0"></use></g></g></svg></span><script type="math/tex">-100 \leqslant x \leqslant 100 </script><span>，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="24.557ex" height="2.227ex" viewBox="0 -755.9 10573.1 958.9" role="img" focusable="false" style="vertical-align: -0.472ex;"><defs><path stroke-width="0" id="E121-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E121-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E121-MJMAIN-2E" d="M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z"></path><path stroke-width="0" id="E121-MJMAIN-31" d="M213 578L200 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153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E121-MJMAIN-2212" x="0" y="0"></use><g transform="translate(778,0)"><use xlink:href="#E121-MJMAIN-30"></use><use xlink:href="#E121-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="778" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="1278" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="1778" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="2278" y="0"></use><use xlink:href="#E121-MJMAIN-31" x="2778" y="0"></use></g><use xlink:href="#E121-MJAMS-2A7D" x="4333" y="0"></use><use xlink:href="#E121-MJMATHI-78" x="5389" y="0"></use><use xlink:href="#E121-MJAMS-2A7D" x="6239" y="0"></use><g transform="translate(7295,0)"><use xlink:href="#E121-MJMAIN-30"></use><use xlink:href="#E121-MJMAIN-2E" x="500" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="778" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="1278" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="1778" y="0"></use><use xlink:href="#E121-MJMAIN-30" x="2278" y="0"></use><use xlink:href="#E121-MJMAIN-31" x="2778" y="0"></use></g></g></svg></span><script type="math/tex">-0.00001 \leqslant x \leqslant 0.00001</script><span>，就显得过大/过小了。</span></p><p><span>另外注意，一旦采用特征缩放，我们就需对所有的输入采用特征缩放，包括训练集、测试集、预测输入等。</span></p><h2><a name="44-梯度下降实践2-学习速率gradient-descent-in-practice-ii---learning-rate" class="md-header-anchor"></a><span>4.4 梯度下降实践2-学习速率(Gradient Descent in Practice II - Learning Rate)</span></h2><p><span>通常，有两种方法来确定函数是否收敛</span></p><ul><li><p><span>多次迭代收敛法</span></p><ul><li><span>无法确定需要多少次迭代</span></li><li><span>较易绘制关于迭代次数的图像</span></li><li><span>根据图像易预测所需的迭代次数</span></li></ul></li><li><p><span>自动化测试收敛法（比较阈值）</span></p><ul><li><span>不易选取阈值</span></li><li><span>代价函数近乎直线时无法确定收敛情况</span></li></ul></li></ul><p><span>对于梯度下降，一般采用多次迭代收敛法来得出最小化代价函数的参数值，自动化测试收敛法（如设定 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.505ex" height="2.928ex" viewBox="0 -956.9 5383.9 1260.5" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E122-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 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571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E122-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E122-MJMAIN-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E122-MJMATHI-4A" x="0" y="0"></use><g transform="translate(799,0)"><use xlink:href="#E122-MJMAIN-28" x="0" y="0"></use><use xlink:href="#E122-MJMATHI-3B8" x="389" y="0"></use><use xlink:href="#E122-MJMAIN-29" x="858" y="0"></use></g><use xlink:href="#E122-MJMAIN-3C" x="2324" y="0"></use><g transform="translate(3380,0)"><use xlink:href="#E122-MJMAIN-31"></use><use xlink:href="#E122-MJMAIN-30" x="500" y="0"></use><g transform="translate(1000,392)"><use transform="scale(0.707)" xlink:href="#E122-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E122-MJMAIN-33" x="778" y="0"></use></g></g></g></svg></span><script type="math/tex">J\left(\theta\right) < {10}^{-3}</script><span> 时判定收敛）则几乎不会被使用。</span></p><p><span>我们可以通过绘制</span><strong><span>代价函数关于迭代次数的图像</span></strong><span>，可视化梯度下降的执行过程，借助直观的图形来发现代价函数趋向于多少时能趋于收敛，依据图像变化情况，确定诸如学习速率的取值，迭代次数的大小等问题。</span></p><p><img src="images/20180108_103357.png" referrerpolicy="no-referrer"></p><p><span>对于学习速率 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span>，一般上图展现的为适中情况，下图中，左图可能表明 </span><strong><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> 过大</span></strong><span>，代价函数</span><strong><span>无法收敛</span></strong><span>，右图可能表明 </span><strong><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> 过小</span></strong><span>，代价函数</span><strong><span>收敛的太慢</span></strong><span>。当然，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> 足够小时，代价函数在每轮迭代后一定会减少。</span></p><p><img src="images/20180108_104701.png" referrerpolicy="no-referrer"></p><p><span>通过不断改变 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script><span> 值，绘制并观察图像，并以此来确定合适的学习速率。 尝试时可取 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path 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type="math/tex">\dots\;0,001,\;0.003,\;0.01,\;0.03,\;0.1,\;\dots</script></p><h2><a name="45-特征和多项式回归features-and-polynomial-regression" class="md-header-anchor"></a><span>4.5 特征和多项式回归(Features and Polynomial Regression)</span></h2><p><span>在特征选取时，我们也可以自己归纳总结，定义一个新的特征，用来</span><strong><span>取代或拆分</span></strong><span>旧的一个或多个特征。比如，对于房屋面积特征来说，我们可以将其拆分为长度和宽度两个特征，反之，我们也可以合并长度和宽度这两个特征为面积这一个特征。</span></p><p><span>线性回归只能以直线来对数据进行拟合，有时候需要使用</span><strong><span>曲线</span></strong><span>来对数据进行拟合，即</span><strong><span>多项式回归(Polynomial Regression)</span></strong><span>。</span></p><p><span>比如一个二次方模型：</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="25.831ex" height="3.044ex" viewBox="0 -906.7 11121.5 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E124-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 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y="0"></use><g transform="translate(9173,0)"><use xlink:href="#E124-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E124-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10095,0)"><use xlink:href="#E124-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E124-MJMAIN-32" x="808" y="487"></use><use transform="scale(0.707)" xlink:href="#E124-MJMAIN-32" x="808" y="-434"></use></g></g></svg></span><script type="math/tex">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}</script></p><p><span>或者三次方模型：</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="33.195ex" height="3.044ex" viewBox="0 -906.7 14292.1 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E125-MJMATHI-68" d="M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 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y="0"></use></g><use xlink:href="#E125-MJMAIN-3D" x="2802" y="0"></use><g transform="translate(3857,0)"><use xlink:href="#E125-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E125-MJMAIN-2B" x="5002" y="0"></use><g transform="translate(6002,0)"><use xlink:href="#E125-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-31" x="663" y="-213"></use></g><g transform="translate(6925,0)"><use xlink:href="#E125-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-31" x="808" y="-213"></use></g><use xlink:href="#E125-MJMAIN-2B" x="8173" y="0"></use><g transform="translate(9173,0)"><use xlink:href="#E125-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-32" x="663" y="-213"></use></g><g transform="translate(10095,0)"><use xlink:href="#E125-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-32" x="808" y="487"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-32" x="808" y="-434"></use></g><use xlink:href="#E125-MJMAIN-2B" x="11343" y="0"></use><g transform="translate(12343,0)"><use xlink:href="#E125-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-33" x="663" y="-213"></use></g><g transform="translate(13266,0)"><use xlink:href="#E125-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-33" x="808" y="509"></use><use transform="scale(0.707)" xlink:href="#E125-MJMAIN-33" x="808" y="-433"></use></g></g></svg></span><script type="math/tex">h_{\theta}\left( x \right)={\theta_{0}}+{\theta_{1}}{x_{1}}+{\theta_{2}}{x_{2}^2}+{\theta_{3}}{x_{3}^3}</script><span> </span></p><p><span>或者平方根模型： </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.129ex" height="3.044ex" viewBox="0 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314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path stroke-width="0" id="E127-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E127-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E127-MJMAIN-31" x="808" y="-213"></use></g></svg></span><script type="math/tex">x_1</script><span> 的范围为 1-1000，那么 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.382ex" height="3.044ex" viewBox="0 -906.7 1025.6 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E128-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 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transform="scale(0.707)" xlink:href="#E129-MJMATHI-6A" x="663" y="-213"></use></g><use xlink:href="#E129-MJMAIN-29" x="1249" y="0"></use></g><use xlink:href="#E129-MJMAIN-3D" x="4085" y="0"></use><use xlink:href="#E129-MJMAIN-30" x="5140" y="0"></use></g></svg></span><script type="math/tex">\frac{\partial}{\partial{\theta_{j}}}J\left( {\theta_{j}} \right)=0</script><span> ，通过解析函数的方式直接计算得出参数向量的值  </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="18.547ex" height="3.518ex" viewBox="0 -1109.9 7985.3 1514.7" role="img" focusable="false" style="vertical-align: -0.94ex;"><defs><path stroke-width="0" id="E130-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 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610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E130-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E130-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E130-MJSZ1-28" d="M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z"></path><path stroke-width="0" id="E130-MJSZ1-29" d="M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 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-142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E130-MJMATHI-3B8" x="0" y="0"></use><use xlink:href="#E130-MJMAIN-3D" x="746" y="0"></use><g transform="translate(1802,0)"><use xlink:href="#E130-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E130-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E130-MJMATHI-54" x="1215" y="513"></use></g><use xlink:href="#E130-MJMATHI-58" x="1915" y="0"></use><use xlink:href="#E130-MJSZ1-29" x="2767" y="-1"></use><g transform="translate(3225,576)"><use transform="scale(0.707)" xlink:href="#E130-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E130-MJMAIN-31" x="778" y="0"></use></g></g><g transform="translate(6031,0)"><use xlink:href="#E130-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E130-MJMATHI-54" x="1215" y="513"></use></g><use xlink:href="#E130-MJMATHI-79" x="7488" y="0"></use></g></svg></span><script type="math/tex">\theta ={{\left( {X^T}X \right)}^{-1}}{X^{T}}y</script><span> ，Octave/Matlab 代码： </span><code>theta = inv(X&#39;*X)*X&#39;*y</code><span>。</span></p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.327ex" height="2.344ex" viewBox="0 -956.9 1862.9 1009.2" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E131-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E131-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E131-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E131-MJMATHI-58" x="0" y="0"></use><g transform="translate(859,409)"><use transform="scale(0.707)" xlink:href="#E131-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E131-MJMAIN-31" x="778" y="0"></use></g></g></svg></span><script type="math/tex">{X}^{-1}</script><span>: 矩阵 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.979ex" height="1.877ex" viewBox="0 -755.9 852 808.1" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E132-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E132-MJMATHI-58" x="0" y="0"></use></g></svg></span><script type="math/tex">X</script><span> 的逆，在 Octave 中，</span><code>inv</code><span> 函数用于计算矩阵的逆，类似的还有 </span><code>pinv</code><span> 函数。</span></p><p><code>X&#39;</code><span>: 在 Octave 中表示矩阵 X 的转置，即 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="3.384ex" height="2.227ex" viewBox="0 -906.7 1457.1 958.9" role="img" focusable="false" style="vertical-align: -0.121ex;"><defs><path stroke-width="0" id="E133-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E133-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E133-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E133-MJMATHI-54" x="1215" y="513"></use></g></svg></span><script type="math/tex">X^T</script></p></blockquote><p><span>下表列出了正规方程法与梯度下降算法的对比</span></p><figure><table><thead><tr><th><span>条件</span></th><th><span>梯度下降</span></th><th><span>正规方程</span></th></tr></thead><tbody><tr><td><span>是否需要选取 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.486ex" height="1.41ex" viewBox="0 -504.6 640 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E76-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E76-MJMATHI-3B1" x="0" y="0"></use></g></svg></span><script type="math/tex">\alpha</script></td><td><span>需要</span></td><td><span>不需要</span></td></tr><tr><td><span>是否需要迭代运算</span></td><td><span>需要</span></td><td><span>不需要</span></td></tr><tr><td><span>特征量大</span><sup class='md-footnote'><a href='#dfref-footnote-1' name='ref-footnote-1'>1</a></sup><span>时</span></td><td><span>适用，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.944ex" height="3.044ex" viewBox="0 -906.7 3420.2 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E134-MJMATHI-4F" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path stroke-width="0" id="E134-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E134-MJMATHI-6B" d="M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"></path><path stroke-width="0" id="E134-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E134-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E134-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E134-MJSZ1-28" d="M152 251Q152 646 388 850H416Q422 844 422 841Q422 837 403 816T357 753T302 649T255 482T236 250Q236 124 255 19T301 -147T356 -251T403 -315T422 -340Q422 -343 416 -349H388Q359 -325 332 -296T271 -213T212 -97T170 56T152 251Z"></path><path stroke-width="0" id="E134-MJSZ1-29" d="M305 251Q305 -145 69 -349H56Q43 -349 39 -347T35 -338Q37 -333 60 -307T108 -239T160 -136T204 27T221 250T204 473T160 636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E134-MJMATHI-4F" x="0" y="0"></use><g transform="translate(929,0)"><use xlink:href="#E134-MJSZ1-28"></use><use xlink:href="#E134-MJMATHI-6B" x="458" y="0"></use><g transform="translate(979,0)"><use xlink:href="#E134-MJMATHI-6E" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E134-MJMAIN-32" x="848" y="513"></use></g><use xlink:href="#E134-MJSZ1-29" x="2032" y="-1"></use></g></g></svg></span><script type="math/tex">O\left(kn^2\right)</script></td><td><span>不适用，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.501ex" height="2.811ex" viewBox="0 -906.7 4090.7 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E135-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E135-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E135-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E135-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 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transform="translate(2698,0)"><use xlink:href="#E135-MJMAIN-29" x="0" y="0"></use><g transform="translate(389,362)"><use transform="scale(0.707)" xlink:href="#E135-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E135-MJMAIN-31" x="778" y="0"></use></g></g></g></svg></span><script type="math/tex">(X^TX)^{-1}</script><span> 复杂度 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="6.734ex" height="3.044ex" viewBox="0 -906.7 2899.2 1310.7" role="img" focusable="false" style="vertical-align: -0.938ex;"><defs><path stroke-width="0" id="E136-MJMATHI-4F" d="M740 435Q740 320 676 213T511 42T304 -22Q207 -22 138 35T51 201Q50 209 50 244Q50 346 98 438T227 601Q351 704 476 704Q514 704 524 703Q621 689 680 617T740 435ZM637 476Q637 565 591 615T476 665Q396 665 322 605Q242 542 200 428T157 216Q157 126 200 73T314 19Q404 19 485 98T608 313Q637 408 637 476Z"></path><path stroke-width="0" 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href='#dfref-footnote-2' name='ref-footnote-2'>2</a></sup></td><td><span>各类模型</span></td><td><span>只适用线性模型，且矩阵需可逆</span></td></tr></tbody></table></figure><p><strong><span>正规方程法的推导过程</span></strong><span>：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n111" cid="n111" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-487-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.954037;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="97.933ex" height="17.636ex" viewBox="0 -4022.7 42165.4 7593.1" role="img" focusable="false" style="vertical-align: -8.293ex; max-width: 100%;"><defs><path stroke-width="0" id="E497-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E497-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E497-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 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transform="translate(3238,0)"><use xlink:href="#E497-MJSZ2-2211" x="0" y="0"></use><g transform="translate(148,-1088)"><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-69" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-3D" x="345" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-31" x="1123" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-6D" x="582" y="1626"></use></g><g transform="translate(4848,0)"><use xlink:href="#E497-MJSZ2-28"></use><g transform="translate(597,0)"><use xlink:href="#E497-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-3B8" x="814" y="-218"></use></g><g transform="translate(1771,0)"><use xlink:href="#E497-MJSZ2-28"></use><g transform="translate(597,0)"><use xlink:href="#E497-MJMATHI-78" x="0" y="0"></use><g transform="translate(572,412)"><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E497-MJSZ2-29" x="2063" y="-1"></use></g><use xlink:href="#E497-MJMAIN-2212" x="4653" y="0"></use><g transform="translate(5653,0)"><use xlink:href="#E497-MJMATHI-79" x="0" y="0"></use><g transform="translate(499,412)"><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-29" x="733" y="0"></use></g></g><use xlink:href="#E497-MJSZ2-29" x="7047" y="-1"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-32" x="10810" y="1239"></use></g></g><g transform="translate(0,-433)"><use xlink:href="#E497-MJMAIN-3D" x="277" y="0"></use><g transform="translate(1055,0)"><g transform="translate(397,0)"><rect stroke="none" width="1498" height="60" x="0" y="220"></rect><use xlink:href="#E497-MJMAIN-31" x="499" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E497-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E497-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xlink:href="#E497-MJMAIN-7C" x="3071" y="0"></use><use xlink:href="#E497-MJMAIN-7C" x="3349" y="0"></use><use xlink:href="#E497-MJMATHI-58" x="3627" y="0"></use><use xlink:href="#E497-MJMATHI-3B8" x="4479" y="0"></use><use xlink:href="#E497-MJMAIN-2212" x="5170" y="0"></use><use xlink:href="#E497-MJMATHI-79" x="6171" y="0"></use><use xlink:href="#E497-MJMAIN-7C" x="6668" y="0"></use><g transform="translate(6946,0)"><use xlink:href="#E497-MJMAIN-7C" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMAIN-32" x="393" y="674"></use></g></g><g transform="translate(0,-2772)"><use xlink:href="#E497-MJMAIN-3D" x="277" y="0"></use><g transform="translate(1055,0)"><g transform="translate(397,0)"><rect stroke="none" width="1498" height="60" x="0" y="220"></rect><use xlink:href="#E497-MJMAIN-31" x="499" y="676"></use><g transform="translate(60,-686)"><use xlink:href="#E497-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E497-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xlink:href="#E497-MJMAIN-28" x="3071" y="0"></use><use xlink:href="#E497-MJMATHI-58" x="3460" y="0"></use><use xlink:href="#E497-MJMATHI-3B8" x="4312" y="0"></use><use xlink:href="#E497-MJMAIN-2212" x="5003" y="0"></use><use xlink:href="#E497-MJMATHI-79" x="6004" y="0"></use><g transform="translate(6501,0)"><use xlink:href="#E497-MJMAIN-29" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E497-MJMATHI-54" x="550" y="583"></use></g><use xlink:href="#E497-MJMAIN-28" x="7487" y="0"></use><use xlink:href="#E497-MJMATHI-58" x="7876" y="0"></use><use xlink:href="#E497-MJMATHI-3B8" x="8728" y="0"></use><use xlink:href="#E497-MJMAIN-2212" x="9420" y="0"></use><use xlink:href="#E497-MJMATHI-79" x="10420" y="0"></use><use xlink:href="#E497-MJMAIN-29" x="10917" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-487">\begin{aligned}
J\left( \theta  \right)& =\frac{1}{2m}\sum\limits_{i=1}^{m}{{{\left( {h_{\theta}}\left( {x^{(i)}} \right)-{y^{(i)}} \right)}^{2}}}\\
& =\frac{1}{2m}||X\theta-y||^2 \\
& =\frac{1}{2m}(X\theta-y)^T(X\theta-y) \hspace{15cm}
\end{aligned}</script></div></div><p><span>展开上式可得</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.472ex" height="3.395ex" viewBox="0 -1007.2 20008.9 1461.5" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E138-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E138-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 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395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E138-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 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585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E138-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E138-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 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636T108 740T60 807T35 839Q35 850 50 850H56H69Q197 743 256 566Q305 425 305 251Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E138-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E138-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E138-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E138-MJMAIN-29" x="1491" y="0"></use><use xlink:href="#E138-MJMAIN-3D" x="2157" y="0"></use><g transform="translate(2935,0)"><g transform="translate(397,0)"><rect stroke="none" width="1094" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E138-MJMAIN-31" x="523" y="571"></use><g transform="translate(59,-376)"><use transform="scale(0.707)" xlink:href="#E138-MJMAIN-32" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-6D" x="500" y="0"></use></g></g></g><g transform="translate(4547,0)"><use xlink:href="#E138-MJSZ1-28"></use><g transform="translate(458,0)"><use xlink:href="#E138-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="663" y="610"></use></g><g transform="translate(1524,0)"><use xlink:href="#E138-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="1215" y="579"></use></g><use xlink:href="#E138-MJMATHI-58" x="2981" y="0"></use><use xlink:href="#E138-MJMATHI-3B8" x="3833" y="0"></use><use xlink:href="#E138-MJMAIN-2212" x="4525" y="0"></use><g transform="translate(5525,0)"><use xlink:href="#E138-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="663" y="610"></use></g><g transform="translate(6592,0)"><use xlink:href="#E138-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="1215" y="579"></use></g><use xlink:href="#E138-MJMATHI-79" x="8049" y="0"></use><use xlink:href="#E138-MJMAIN-2212" x="8768" y="0"></use><g transform="translate(9768,0)"><use xlink:href="#E138-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="705" y="513"></use></g><use xlink:href="#E138-MJMATHI-58" x="10865" y="0"></use><use xlink:href="#E138-MJMATHI-3B8" x="11717" y="0"></use><use xlink:href="#E138-MJMAIN-2B" x="12408" y="0"></use><g transform="translate(13408,0)"><use xlink:href="#E138-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E138-MJMATHI-54" x="705" y="513"></use></g><use xlink:href="#E138-MJMATHI-79" x="14505" y="0"></use><use xlink:href="#E138-MJSZ1-29" x="15002" y="-1"></use></g></g></svg></span><script type="math/tex">J(\theta )= \frac{1}{2m}\left( {{\theta }^{T}}{{X}^{T}}X\theta -{{\theta}^{T}}{{X}^{T}}y-{{y}^{T}}X\theta + {{y}^{T}}y \right)</script></p><p><span>注意到 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.016ex" height="3.044ex" viewBox="0 -1007.2 3020.9 1310.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E139-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E139-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E139-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E139-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E139-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E139-MJMATHI-54" x="663" y="610"></use><g transform="translate(1066,0)"><use xlink:href="#E139-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E139-MJMATHI-54" x="1215" y="579"></use></g><use xlink:href="#E139-MJMATHI-79" x="2523" y="0"></use></g></svg></span><script type="math/tex">{{\theta}^{T}}{{X}^{T}}y</script><span> 与 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.616ex" height="2.811ex" viewBox="0 -906.7 2418 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E140-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E140-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E140-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E140-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E140-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E140-MJMATHI-54" x="705" y="513"></use><use xlink:href="#E140-MJMATHI-58" x="1096" y="0"></use><use xlink:href="#E140-MJMATHI-3B8" x="1948" y="0"></use></g></svg></span><script type="math/tex">{{y}^{T}}X\theta</script><span> 都为标量，实际上是等价的，则：</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="35.865ex" height="3.278ex" viewBox="0 -956.9 15441.7 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E141-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E141-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E141-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E141-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E141-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E141-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E141-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E141-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E141-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path 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0Z"></path><path stroke-width="0" id="E141-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E141-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E141-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E141-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E141-MJMAIN-5D" d="M22 710V750H159V-250H22V-210H119V710H22Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E141-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E141-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E141-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E141-MJMAIN-29" x="1491" y="0"></use><use xlink:href="#E141-MJMAIN-3D" x="2157" y="0"></use><g transform="translate(2935,0)"><g transform="translate(397,0)"><rect stroke="none" width="1094" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E141-MJMAIN-31" x="523" y="571"></use><g transform="translate(59,-376)"><use transform="scale(0.707)" xlink:href="#E141-MJMAIN-32" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-6D" x="500" y="0"></use></g></g></g><use xlink:href="#E141-MJMAIN-5B" x="4547" y="0"></use><g transform="translate(4825,0)"><use xlink:href="#E141-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-54" x="1215" y="513"></use></g><use xlink:href="#E141-MJMATHI-58" x="6283" y="0"></use><use xlink:href="#E141-MJMATHI-3B8" x="7135" y="0"></use><use xlink:href="#E141-MJMAIN-2212" x="7826" y="0"></use><use xlink:href="#E141-MJMAIN-32" x="8826" y="0"></use><g transform="translate(9326,0)"><use xlink:href="#E141-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-54" x="663" y="513"></use></g><g transform="translate(10393,0)"><use xlink:href="#E141-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-54" x="1215" y="513"></use></g><use xlink:href="#E141-MJMATHI-79" x="11850" y="0"></use><use xlink:href="#E141-MJMAIN-2B" x="12569" y="0"></use><g transform="translate(13569,0)"><use xlink:href="#E141-MJMATHI-79" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E141-MJMATHI-54" x="705" y="513"></use></g><use xlink:href="#E141-MJMATHI-79" x="14666" y="0"></use><use xlink:href="#E141-MJMAIN-5D" x="15163" y="0"></use></g></svg></span><script type="math/tex">J(\theta) = \frac{1}{2m}[X^TX\theta-2\theta^TX^Ty+y^Ty]</script></p><p><span>接下来对</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.366ex" height="2.577ex" viewBox="0 -806.1 1880 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E142-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E142-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E142-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E142-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E142-MJMATHI-4A" x="0" y="0"></use><use xlink:href="#E142-MJMAIN-28" x="633" y="0"></use><use xlink:href="#E142-MJMATHI-3B8" x="1022" y="0"></use><use xlink:href="#E142-MJMAIN-29" x="1491" y="0"></use></g></svg></span><script type="math/tex">J(\theta )</script><span> 求偏导，根据矩阵的求导法则:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="21.343ex" height="3.628ex" viewBox="0 -1107.7 9189.4 1562" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E143-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E143-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E143-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E143-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E143-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E143-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E143-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E143-MJMAIN-54" d="M36 443Q37 448 46 558T55 671V677H666V671Q667 666 676 556T685 443V437H645V443Q645 445 642 478T631 544T610 593Q593 614 555 625Q534 630 478 630H451H443Q417 630 414 618Q413 616 413 339V63Q420 53 439 50T528 46H558V0H545L361 3Q186 1 177 0H164V46H194Q264 46 283 49T309 63V339V550Q309 620 304 625T271 630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></path><path stroke-width="0" id="E143-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2652" height="60" x="0" y="220"></rect><g transform="translate(60,411)"><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-64" x="0" y="0"></use><g transform="translate(369,0)"><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E143-MJMATHI-54" x="1215" y="579"></use></g><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-41" x="1980" y="0"></use><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-58" x="2730" y="0"></use></g><g transform="translate(840,-395)"><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E143-MJMATHI-58" x="523" y="0"></use></g></g><use xlink:href="#E143-MJMAIN-3D" x="3170" y="0"></use><use xlink:href="#E143-MJMAIN-28" x="4226" y="0"></use><use xlink:href="#E143-MJMATHI-41" x="4615" y="0"></use><use xlink:href="#E143-MJMAIN-2B" x="5587" y="0"></use><g transform="translate(6587,0)"><use xlink:href="#E143-MJMATHI-41" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E143-MJMAIN-54" x="1060" y="513"></use></g><use xlink:href="#E143-MJMAIN-29" x="7948" y="0"></use><use xlink:href="#E143-MJMATHI-58" x="8337" y="0"></use></g></svg></span><script type="math/tex">\frac{dX^TAX}{dX}=(A+A^\mathrm{T})X</script></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="10.159ex" height="3.628ex" viewBox="0 -1107.7 4374 1562" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E144-MJMATHI-64" d="M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E144-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E144-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path><path stroke-width="0" id="E144-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path><path stroke-width="0" id="E144-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="2050" height="60" x="0" y="220"></rect><g transform="translate(60,411)"><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-64" x="0" y="0"></use><g transform="translate(369,0)"><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.5)" xlink:href="#E144-MJMATHI-54" x="1215" y="579"></use></g><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-41" x="1980" y="0"></use></g><g transform="translate(539,-395)"><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E144-MJMATHI-58" x="523" y="0"></use></g></g><use xlink:href="#E144-MJMAIN-3D" x="2568" y="0"></use><use xlink:href="#E144-MJMATHI-41" x="3624" y="0"></use></g></svg></span><script type="math/tex">\frac{dX^TA}{dX}={A}</script></p><p><span>所以有:</span></p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="46.259ex" height="3.978ex" viewBox="0 -1208.2 19916.9 1712.8" role="img" focusable="false" style="vertical-align: -1.172ex;"><defs><path stroke-width="0" id="E145-MJMAIN-2202" d="M202 508Q179 508 169 520T158 547Q158 557 164 577T185 624T230 675T301 710L333 715H345Q378 715 384 714Q447 703 489 661T549 568T566 457Q566 362 519 240T402 53Q321 -22 223 -22Q123 -22 73 56Q42 102 42 148V159Q42 276 129 370T322 465Q383 465 414 434T455 367L458 378Q478 461 478 515Q478 603 437 639T344 676Q266 676 223 612Q264 606 264 572Q264 547 246 528T202 508ZM430 306Q430 372 401 400T333 428Q270 428 222 382Q197 354 183 323T150 221Q132 149 132 116Q132 21 232 21Q244 21 250 22Q327 35 374 112Q389 137 409 196T430 306Z"></path><path stroke-width="0" id="E145-MJMATHI-4A" d="M447 625Q447 637 354 637H329Q323 642 323 645T325 664Q329 677 335 683H352Q393 681 498 681Q541 681 568 681T605 682T619 682Q633 682 633 672Q633 670 630 658Q626 642 623 640T604 637Q552 637 545 623Q541 610 483 376Q420 128 419 127Q397 64 333 21T195 -22Q137 -22 97 8T57 88Q57 130 80 152T132 174Q177 174 182 130Q182 98 164 80T123 56Q115 54 115 53T122 44Q148 15 197 15Q235 15 271 47T324 130Q328 142 387 380T447 625Z"></path><path stroke-width="0" id="E145-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E145-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E145-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E145-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E145-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E145-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E145-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E145-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 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y="674"></use></g></g></svg></span><script type="math/tex">{x_{1}}={x_{2}}*{{\left( 3.28 \right)}^{2}}</script><span>。</span></p></li><li><p><span>特征数量大于训练集的数量</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="8.337ex" height="2.577ex" viewBox="0 -806.1 3589.6 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E148-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E148-MJMATHI-6D" d="M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E148-MJAMS-2A7D" d="M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM94 170Q102 172 104 172Q110 171 254 103T535 -30T678 -98Q694 -106 694 -118Q694 -136 676 -138H670L382 -2Q92 135 90 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682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z"></path><path stroke-width="0" id="E149-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E149-MJMATHI-58" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E149-MJMATHI-54" x="1215" y="513"></use><use xlink:href="#E149-MJMATHI-58" x="1457" y="0"></use></g></svg></span><script type="math/tex">X^TX</script><span> 的结果不可逆，可尝试：</span></p><ul><li><span>减少多余/重复特征</span></li><li><span>增加训练集数量</span></li><li><span>使用正则化（后文）</span></li></ul><p><span>对于这类不可逆的矩阵，我们称之为</span><strong><span>奇异矩阵</span></strong><span>或</span><strong><span>退化矩阵</span></strong><span>。</span></p><p><span>这种情况下，如果还想使用正规方程法，在Octave中，可以选用 </span><code>pinv</code><span> 函数，</span><code>pinv</code><span> 区别于 </span><code>inv</code><span>，</span><code>pinv</code><span> 函数被称为伪逆函数，在矩阵不可逆的时候，使用这个函数仍可正确地计算出 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E74-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E74-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 的值。</span></p><h1><a name="5-octavematlab-tutorial" class="md-header-anchor"></a><span>5 Octave/Matlab Tutorial</span></h1><p><span>复习时可直接倍速回顾视频，笔记整理暂留。</span></p><h2><a name="51-basic-operations" class="md-header-anchor"></a><span>5.1 Basic Operations</span></h2><h2><a name="52-moving-data-around" class="md-header-anchor"></a><span>5.2 Moving Data Around</span></h2><h2><a name="53-computing-on-data" class="md-header-anchor"></a><span>5.3 Computing on Data</span></h2><h2><a name="54-plotting-data" class="md-header-anchor"></a><span>5.4 Plotting Data</span></h2><h2><a name="55-control-statements-for-while-if-statement" class="md-header-anchor"></a><span>5.5 Control Statements: for, while, if statement</span></h2><h2><a name="56-向量化vectorization" class="md-header-anchor"></a><span>5.6 向量化(Vectorization)</span></h2><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n152" cid="n152" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-489-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="14.88ex" height="7.13ex" viewBox="0 -1660.6 6406.7 3069.8" role="img" focusable="false" style="vertical-align: -3.273ex; max-width: 100%;"><defs><path stroke-width="0" id="E499-MJSZ2-2211" d="M60 948Q63 950 665 950H1267L1325 815Q1384 677 1388 669H1348L1341 683Q1320 724 1285 761Q1235 809 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345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E499-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E499-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 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445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E499-MJSZ2-2211" x="0" y="0"></use><g transform="translate(124,-1088)"><use transform="scale(0.707)" xlink:href="#E499-MJMATHI-6A" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E499-MJMAIN-3D" x="412" y="0"></use><use transform="scale(0.707)" xlink:href="#E499-MJMAIN-30" x="1189" y="0"></use></g><use transform="scale(0.707)" xlink:href="#E499-MJMATHI-6E" x="721" y="1626"></use><g transform="translate(1610,0)"><use xlink:href="#E499-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E499-MJMATHI-6A" x="663" y="-213"></use></g><g transform="translate(2470,0)"><use xlink:href="#E499-MJMATHI-78" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E499-MJMATHI-6A" x="808" y="-213"></use></g><use xlink:href="#E499-MJMAIN-3D" x="3712" y="0"></use><g transform="translate(4767,0)"><use xlink:href="#E499-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E499-MJMATHI-54" x="663" y="583"></use></g><use xlink:href="#E499-MJMATHI-78" x="5834" y="0"></use></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-489">\sum\limits_{j=0}^n\theta_jx_j=\theta^Tx</script></div></div><h2><a name="5x-常用函数整理" class="md-header-anchor"></a><span>5.x 常用函数整理</span></h2><div class='footnotes-area'  ><hr/>
<div class='footnote-line'><span class='md-fn-count'>1</span> <span>一般来说，当 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.394ex" height="1.435ex" viewBox="0 -504.2 600 617.7" role="img" focusable="false" style="vertical-align: -0.264ex;"><defs><path stroke-width="0" id="E137-MJMATHI-6E" d="M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E137-MJMATHI-6E" x="0" y="0"></use></g></svg></span><script type="math/tex">n</script><span> 超过 10000 时，对于正规方程而言，特征量较大。</span> <a name='dfref-footnote-1' href='#ref-footnote-1' title='回到文档' class='reversefootnote' >↩</a></div>
<div class='footnote-line'><span class='md-fn-count'>2</span> <span>梯度下降算法的普适性好，而对于特定的线性回归模型，正规方程是很好的替代品。</span> <a name='dfref-footnote-2' href='#ref-footnote-2' title='回到文档' class='reversefootnote' >↩</a></div></div></div>
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